Quantum light propagation in longitudinally inhomogeneous waveguides as a spatial Lewis-Ermakov physical invariance

We study the propagation of quantum states of light in separable longitudinally inhomogeneous waveguides. By means of the usual quantization approach this kind of media would lead to the unphysical result of quantum noise squeezing. This problem is solved by means of generalized canonical transformations in a comoving frame. Under these transformations the generator of propagation is a Lewis-Ermakov invariant in space which is quantized and, accordingly, a propagator consistent with experiments is obtained. Finally, we show that the net effect produced by propagation in these media is a quantum Gouy's phase with applications in quantum information processing and sensing.